4.1 Introduction

4.1.1 The Intent of Content Markup

As has been noted in the introductory section of this recommendation, mathematics can be distinguished by its use of a (relatively) formal language, mathematical notation. However, mathematics and its presentation should not be viewed as one and the same thing. Mathematical sums or products exist and are meaningful to many applications completely without regard to how they are rendered aurally or visually. The intent of the content markup in the Mathematical Markup Language is to provide an explicit encoding of the
underlying mathematical structure of an expression, rather than any particular rendering for the expression.

There are many reasons for providing a specific encoding for content. Even a disciplined and systematic use of presentation tags cannot properly capture this semantic information. This is because without additional information it is impossible to decide if a particular presentation was chosen deliberately to encode the mathematical structure or simply to achieve a particular visual or aural effect. Furthermore, an author using the same encoding to deal with both the presentation and mathematical structure might find a particular presentation encoding unavailable simply because convention had reserved it for a different semantic meaning.

The difficulties stem from the fact that there are many to one mappings from presentation to semantics and vice versa. For example the mathematical construct
`
H multiplied by
e' is often encoded using an explicit operator as in
H ×
e. In different presentational contexts, the multiplication operator might be invisible
`
He', or rendered as the spoken word
`times'. Generally, many different presentations are possible depending on the context and style preferences of the author or reader. Thus, given
`
He' out of context it may be impossible to decide if this is the name of a chemical or a mathematical product of two variables
H and
e.

Mathematical presentation also changes with culture and time: some expressions in combinatorial mathematics today have one meaning to an Russian mathematician, and quite another to a French mathematician; see
Section 5.4.1 [Notational Style Sheets] for an example. Notations may lose currency, for example the use of musical sharp and flat symbols to denote maxima and minima
[Chaundy1954]. A notation in use in 1644 for the multiplication mentioned above was
He
[Cajori1928].

When we encode the underlying mathematical structure explicitly, without regard to how it is presented aurally or visually, we are able to interchange information more precisely with those systems that are able to manipulate the mathematics. In the trivial example above, such a system could substitute values for the variables
H and
e and evaluate the result. Further interesting application areas include interactive textbooks and other teaching aids.

4.1.2 The Scope of Content Markup

The semantics of general mathematical notation is not a matter of consensus. It would be an enormous job to systematically codify most of mathematics - a task that can never be complete. Instead, MathML makes explicit a relatively small number of commonplace mathematical constructs, chosen carefully to be sufficient in a large number of applications. In addition, it provides a mechanism for associating semantics with new notational constructs. In this way, mathematical concepts that are not in the base collection of elements can still be encoded (Section 4.2.6 [Syntax and Semantics]).

The base set of content elements are chosen to be adequate for simple coding of most of the formulas used from kindergarten to the end of high school in the United States, and probably beyond through the first two years of college, that is up to A-Level or Baccalaureate level in Europe. Subject areas covered to some extent in MathML are:

arithmetic, algebra, logic and relations

calculus and vector calculus

set theory

sequences and series

elementary classical functions

statistics

linear algebra

It is not claimed, or even suggested, that the proposed set of elements is complete for these areas, but the provision for author extensibility greatly alleviates any problem omissions from this finite list might cause.

4.1.3 Basic Concepts of Content Markup

The design of the MathML content elements are driven by the following principles:

The expression tree structure of a mathematical expression should be directly encoded by the MathML content elements.

The encoding of an expression tree should be explicit, and not dependent on the special parsing of
PCDATA or on additional processing such as operator precedence parsing.

The basic set of mathematical content constructs that are provided should have default mathematical semantics.

There should be a mechanism for associating specific mathematical semantics with the constructs.

The primary goal of the content encoding is to establish explicit connections between mathematical structures and their mathematical meanings. The content elements correspond directly to parts of the underlying mathematical expression tree. Each structure has an associated default semantics and there is a mechanism for associating new mathematical definitions with new constructs.

Significant advantages to the introduction of content-specific tags include:

Usage of presentation elements is less constrained. When mathematical semantics are inferred from presentation markup, processing agents must either be quite sophisticated, or they run the risk of inferring incomplete or incorrect semantics when irregular constructions are used to achieve a particular aural or visual effect.

It is immediately clear which kind of information is being encoded simply by the kind of elements that are used.

Combinations of semantic and presentation elements can be used to convey both the appearance and its mathematical meaning much more effectively than simply trying to infer one from the other.

Expressions described in terms of content elements must still be rendered. For common expressions, default visual presentations are usually clear.
`Take care of the sense and the sounds will take care of themselves' wrote Lewis Carroll
[Carroll1871]. Default presentations are included in the detailed description of each element occurring in
Section 4.4 [The Content Markup Elements].

To accomplish these goals, the MathML content encoding is based on the concept of an expression tree. A content expression tree is constructed from a collection of more primitive objects, referred to herein as
containers and
operators. MathML possesses a rich set of predefined container and operator objects, as well as constructs for combining containers and operators in mathematically meaningful ways. The syntax and usage of these content elements and constructions is described in the next section.

4.2 Content Element Usage Guide

Since the intent of MathML content markup is to encode mathematical expressions in such a way that the mathematical structure of the expression is clear, the syntax and usage of content markup must be consistent enough to facilitate automated semantic interpretation. There must be no doubt when, for example, an actual sum, product or function application is intended and if specific numbers are present, there must be enough information present to reconstruct the correct number for purposes of computation. Of course, it is still up to a MathML-compliant processor to decide what is to be done with such a content-based expression, and computation is only one of many options. A renderer or a structured editor might simply use the data and its own built-in knowledge of mathematical structure to render the object. Alternatively, it might manipulate the object to build a new mathematical object. A more computationally oriented system might attempt to carry out the indicated operation or function evaluation.

The purpose of this section is to describe the intended, consistent usage. The requirements involve more than just satisfying the syntactic structure specified by an XML DTD. Failure to conform to the usage as described below will result in a MathML error, even though the expression may be syntactically valid according to the DTD.

4.2.1 Overview of Syntax and Usage

MathML content encoding is based on the concept of an expression tree. As a general rule, the terminal nodes in the tree represent basic mathematical objects, such as numbers, variables, arithmetic operations and so on. The internal nodes in the tree generally represent some kind of function application or other mathematical construction that builds up a compound object. Function application provides the most important example; an internal node might represent the application of a function to several arguments, which are themselves represented by the terminal nodes underneath the internal node.

The MathML content elements can be grouped into the following categories based on their usage:

containers

operators and functions

qualifiers

relations

conditions

semantic mappings

constants and symbols

These are the building blocks out of which MathML content expressions are constructed. Each category is discussed in a separate section below. In the remainder of this section, we will briefly introduce some of the most common elements of each type, and consider the general constructions for combining them in mathematically meaningful ways.

4.2.1.1 Constructing Mathematical Objects

Content expression trees are built up from basic mathematical objects. At the lowest level,
leaf nodes are encapsulated in non-empty elements that define their type. Numbers and symbols are marked by the
token elements
cn and
ci. More elaborate constructs such as sets, vectors and matrices are also marked using elements to denote their types, but rather than containing data directly, these
container elements are constructed out of other elements. Elements are used in order to clearly identify the underlying objects. In this way, standard XML parsing can be used and attributes can be used to specify global properties of the objects.

The containers such as
<cn>12345<cn/> ,
<ci>x</ci> and
<csymbol definitionURL="mySymbol.htm" encoding="text">S</csymbol>represent mathematical numbers , identifiers and externally defined symbols. Below, we will look at
operator elements such as
plus or
sin, which provide access to the basic mathematical operations and functions applicable to those objects. Additional containers such as
set for sets, and
matrix for matrices are provided for representing a variety of common compound objects.

For example, the number 12345 is encoded as

<cn>12345</cn>

The attributes and
PCDATA content together provide the data necessary for an application to parse the number. For example, a default base of 10 is assumed, but to communicate that the underlying data was actually written in base 8, simply set the
base attribute to 8 as in

<cn base="8">12345</cn>

while the complex number 3 + 4i can be encoded as

<cn type="complex">3<sep/>4</cn>

Such information makes it possible for another application to easily parse this into the correct number.

This invokes default semantics associated with the
vector element, namely an arbitrary element of a finite-dimensional vector space.

By using the
ci and
csymbol elements we have made clear that we are referring to a mathematical identifier or symbol but this does not say anything about how it should be rendered. By default a symbol is rendered as if the
ci or
csymbolelement were actually the presentation element
mi (see
Section 3.2.2 [Identifier (mi)]). The actual rendering of a mathematical symbol can be made as elaborate as necessary simply by using the more elaborate presentational constructs (as described in
Chapter 3 [Presentation Markup]) in the body of the
ci or
csymbol element.

The default rendering of a simple
cn-tagged object is the same as for the presentation element
mn with some provision for overriding the presentation of the
PCDATA by providing explicit
mntags. This is described in detail in
Section 4.4 [The Content Markup Elements].

The issues for compound objects such as sets, vectors and matrices are all similar to those outlined above for numbers and symbols. Each such object has global properties as a mathematical object that impact how they are to be parsed. This may affect everything from the interpretation of operations that are applied to them through to how to render the symbols representing them. These mathematical properties are captured by setting attribute values.

4.2.1.2 Constructing General Expressions

The notion of constructing a general expression tree is essentially that of applying an operator to sub-objects. For example, the sum
a +
b can be thought of as an application of the addition operator to two arguments
a and
b. In MathML, elements are used for operators for much the same reason that elements are used to contain objects. They are recognized at the level of XML parsing, and their attributes can be used to record or modify the intended semantics. For example, with the MathML
plus element, setting the
definitionURL and
encodingattributes as in

There is also another reason for using elements to denote operators. There is a crucial semantic distinction between the function itself and the expression resulting from applying that function to zero or more arguments which must be captured. This is addressed by making the functions self-contained objects with their own properties and providing an explicit
apply construct corresponding to function application. We will consider the
apply construct in the next section.

MathML contains many pre-defined operator elements, covering a range of mathematical subjects. However, an important class of expressions involve unknown or user-defined functions and symbols. For these situations, MathML provides a general
csymbol element, which is discussed below.

4.2.1.3 The
apply construct

The most fundamental way of building up a mathematical expression in MathML content markup is the
apply construct. An
apply element typically applies an operator to its arguments. It corresponds to a complete mathematical expression. Roughly speaking, this means a piece of mathematics that could be surrounded by parentheses or
`logical brackets' without changing its meaning.

For example, (x +
y) might be encoded as

<apply>
<plus/>
<ci> x </ci>
<ci> y </ci>
</apply>

The opening and closing tags of
apply specify exactly the scope of any operator or function. The most typical way of using
apply is simple and recursive. Symbolically, the content model can the described as:

<apply>
opab </apply>

where the
operands a and b are containers or other content-based elements themselves, and
op is an operator or function. Note that since
apply is a container, this allows
apply constructs to be nested to arbitrary depth.

An
apply may in principle have any number of operands:

<apply> op a b [c...] <apply>

For example, (x +
y +
z) can be encoded as

<apply>
<plus/>
<ci> a </ci>
<ci> b </ci>
<ci> c </ci>
</apply>

Mathematical expressions involving a mixture of operations result in nested occurrences of
apply. For example,
ax +
b would be encoded as

There is no need to introduce parentheses or to resort to operator precedence in order to parse the expression correctly. The
apply tags provide the proper grouping for the re-use of the expressions within other constructs. Any expression enclosed by an
apply element is viewed as a single coherent object.

Both the function and the arguments may be simple identifiers or more complicated expressions.

In MathML 1.0 , another construction closely related to the use of the
apply element with operators and arguments was the
reln element. The
relnelement was used to denote that a mathematical relation holds between its arguments, as opposed to applying an operator. Thus, the MathML markup for the expression
x <
y was given in MathML 1.0 by:

<reln>
<lt/>
<ci> x </ci>
<ci> y </ci>
</reln>

In MathML 2.0, the
apply construct is used with all operators, including logical operators. The expression above becomes

<apply>
<lt/>
<ci> x </ci>
<ci> y </ci>
</apply>

in MathML 2.0. The use of
reln with relational operators is supported
for reasons of backwards compatibility, but deprecated. Authors creating new content are
encouraged to use
apply in all cases.

4.2.1.4 Explicitly defined functions and operators

The most common operations and functions such as
plus and
sin have been predefined explicitly as empty elements (see
Section 4.4 [The Content Markup Elements]). They have
type and
definitionURLattributes, and by changing these attributes, the author can record that a different sort of algebraic operation is intended. This allows essentially the same notation to be re-used for a discussion taking place in a different algebraic domain.

Due to the nature of mathematics the notation must be extensible. The key to extensibility is the ability of the user to define new functions and other symbols to expand the terrain of mathematical discourse.

It is always possible to create arbitrary expressions, and then to use them as symbols in the language. Their properties can then be inferred directly from that usage as was done in the previous section. However, such an approach would preclude being able to encode the fact that the construct was a known symbol, or to record its mathematical properties except by actually using it. The
csymbol element is used as a container to construct a new symbol in much the same way that
ci is used to construct an identifier. (Note that
`symbol' is used here in the abstract sense and has no connection with any presentation of the construct on screen or paper). The difference in usage is that
csymbol should refer to some mathematically defined concept with an external definition referenced via the
definitionURL attribute, whereas
ci is used for identifiers that are essentially
`local' to the MathML expression and do not use any external definition mechanism. The target of the
definitionURLattribute on the
csymbol element may encode the definition in any format: the particular encoding in use is given by the
encoding attribute

To use
csymbol to describe a completely new function, we write for example

The
definitionURL attribute specifies a URI that provides a written definition for the
Christoffel symbol. Suggested default definitions for the content elements of MathML appear in
Appendix C [Content Element Definitions] in a format based on OpenMath, although there is no requirement that a particular format be used. The role of the
definitionURL attribute is very similar to the role of definitions included at the beginning of many mathematical papers, and which often just refer to a definition used by a particular book.

MathML 1.0 supported the use of the
fn to encode the fact that a construct is explicitly being used as a function or operator. To record the fact that
F+
G is being used semantically as if it were a function, it was encoded as:

<fn>
<apply>
<plus/>
<ci>F</ci>
<ci>G</ci>
</apply>
</fn>

This usage, although allowed in MathML 2.0 for reasons of backwards compatibility,
is now deprecated.
The fact that a construct is being used as an operator is clear from the position of the construct as the
first child of the
apply. If it is required to add additional information to the construct, it should be wrapped in a
semanticselement, for example:

MathML 1.0 supported the use of
definitionURL with
fn to refer to external definitions for user-defined
functions. This usage, although allowed for reasons of backwards
compatibility, is deprecated in
MathML 2.0 in favour of using
csymbol to define the function, and then
apply to link the function to its arguments. For example:

4.2.1.5 The inverse construct

Given functions, it is natural to have functional inverses. This is handled by the
inverse element.

Functional inverses can be problematic from a mathematical point of view in that it implicitly involves the definition of an inverse for an arbitrary function
F. Even at the K-through-12 level the concept of an inverse
F-1 of many common functions
Fis not used in a uniform way. For example, the definitions used for the inverse trigonometric functions may differ slightly depending on the choice of domain and/or branch cuts.

MathML adopts the view: if
F is a function from a domain
D to
D', then the inverse
G of
F is a function over
D' such that
G(F(x)) =
x for
x in
D. This definition does not assert that such an inverse exists for all or indeed any
x in
D, or that it is single-valued anywhere. Also, depending on the functions involved, additional properties such as
F(G(y)) =
y for
y in
D' may hold.

The
inverse element is applied to a function whenever an inverse is required. For example, application of the inverse sine function to
x, i.e. sin
-1 (x), is encoded as:

<apply>
<apply> <inverse/> <sin/> </apply>
<ci> x </ci>
</apply>

While
arcsin is one of the predefined MathML functions, an explicit reference to sin
-1(x) might occur in a document discussing possible definitions of
arcsin.

4.2.1.6 The declare construct

Consider a document discussing the vectors
A = (a, b,
c) and
B = (d, e,
f), and later including the expression
V =
A +
B. It is important to be able to communicate the fact that wherever
A and
Bare used they represent a particular vector. The properties of that vector may determine aspects of operators such as
plus.

The simple fact that
A is a vector can be communicated by using the markup

<ci type="vector">A</ci>

but this still does not communicate, for example, which vector is involved
or its dimensions.

The declare construct is used to associate
specific properties or meanings with an object. The actual declaration
itself is not rendered visually (or in any other form). However, it
indirectly impacts the semantics of all affected uses of the declared
object.

The scope of a declaration is, by default, local to the MathML element
in which the declaration is made. If the scopeattribute of the declare
element is set to global, the declaration applies to
the entire MathML expression in which it appears.

The uses of the declare element range from
resetting default attribute values to associating an expression with a
particular instance of a more elaborate structure. Subsequent uses of the
original expression (within the scope of the declare) play the same semantic role as would the
paired object.

remains unchanged but the expression can be interpreted properly as vector addition.

There is no requirement to declare an expression to stand for a specific object. For example, the declaration

<declare type="vector">
<ci> A </ci>
</declare>

specifies that
A is a vector without indicating the number of components or the values of specific components. The possible values for the
type attribute include all the predefined container element names such as
vector,
matrix or
set (see
Section 4.3.2.9 [type]).

4.2.1.7 The lambda construct

The lambda calculus allows a user to construct a function from a variable and an expression. For example, the lambda construct underlies the common mathematical idiom illustrated here:

Let
f be the function taking
x to
x2 + 2

There are various notations for this concept in mathematical literature, such as
(x,
F(x)) = F or
(x,
[F]) =F, where x is a free variable in F.

This concept is implemented in MathML with the lambda element. A lambda construct with n
internal variables is encoded by a lambda element
with n+1 children. All but the last child must be bvar elements containing the identifiers of the
internal variables. The last child is an expression defining the
function. This is typically an apply, but can also
be any container element.

4.2.1.8 The use of qualifier elements and the condition construct

The last example of the preceding section illustrates the use of
qualifier elements
lowlimit,
uplimit, and
bvar used in conjunction with the
int element. A number of common mathematical constructions involve additional data that is either implicit in conventional notation, such as a bound variable, or thought of as part of the operator rather than an argument, as is the case with the limits of a definite integral.

Content markup uses qualifier elements in conjunction with a number of operators, including integrals, sums, series, and certain differential operators. Qualifier elements appear in the same
apply element with one of these operators. In general, they must appear in a certain order, and their precise meaning depends on the operators being used. For details, see
Section 4.2.3.2 [Operators taking Qualifiers].

The qualifier element
bvar is also used in another important MathML construction. The
condition element is used to place conditions on bound variables in other expressions. This allows MathML to define sets by rule, rather than enumeration, for example. The following markup, for instance, encodes the set {
x |
x < 1}:

4.2.1.9 Rendering of Content elements

While the primary role of the MathML content element set is to directly encode the mathematical structure of expressions independent of the notation used to present the objects, rendering issues cannot be ignored. Each content element has a default rendering, given in
Section 4.4 [The Content Markup Elements], and several mechanisms (including
Section 4.3.3.2 [General Attributes]) are provided for associating a particular rendering with an object.

4.2.2 Containers

Containers provide a means for the construction of mathematical objects of a given type.

The
cn element is the MathML token element used to represent numbers. The supported types of numbers include:
real,
integer,
rational,
complex-cartesian, and
complex-polar, with
real being the default type. An attribute
base (with default value
10) is used to help specify how the content is to be parsed. The content itself is essentially
PCDATA, separated by
<sep/> when two parts are needed in order to fully describe a number. For example, the real number 3 is constructed by
<cn type="real"> 3 </cn>, while the rational number 3/4 is constructed as
<cn type="rational"> 3<sep/>4 </cn>. The detailed structure and specifications are provided in
Section 4.4.1.1 [Number (cn)].

ci

The
ci element, or
`content identifier' is used to construct a variable, or an identifier. A
type attribute indicates the type of object the symbol represents. Typically,
ci represents a real scalar, but no default is specified. The content is either
PCDATA or a general presentation construct (see
Section 3.1.5 [Summary of Presentation Elements]). For example,

<ci>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</ci>

encodes an atomic symbol that displays visually as
c1which, for purposes of content, is treated as a single symbol representing a real number. The detailed structure and specifications is provided in
Section 4.4.1.2 [Identifier (ci)].

csymbol

The
csymbol element, or
`content symbol' is used to construct a symbol whose semantics are not part of the core content elements provided by MathML, but defined externally.
csymbol does not make any attempt to describe how to map the arguments occurring in any application of the function into a new MathML expression. Instead, it depends on its
definitionURL attribute to point to a particular meaning, and the
encoding attribute to give the syntax of this definition. The content of a
csymbol is either
PCDATAor a general presentation construct (see
Section 3.1.5 [Summary of Presentation Elements]). For example,

encodes an atomic symbol that displays visually as
C2 and that, for purposes of content, is treated as a single symbol representing the space of twice-differentiable continuous functions. The detailed structure and specifications is provided in
Section 4.4.1.3 [Externally defined symbol (csymbol)].

4.2.2.2 Constructors

MathML provides a number of elements for combining elements into familiar compound objects. The compound objects include things like lists, sets. Each constructor produces a new type of object.

interval

The
interval element is described in detail in
Section 4.4.2.4 [Interval (interval)]. It denotes an interval on the real line with the values represented by its children as end points. The
closure attribute is used to qualify the type of interval being represented. For example,

The
set and
list elements are described in detail in
Section 4.4.6.1 [Set (set)] and
Section 4.4.6.2 [List (list)]. Typically, the child elements of a possibly empty
list element are the actual components of an ordered
list. For example, an ordered list of the three symbols
a,
b, and
c is encoded as

<list> <ci> a </ci> <ci> b </ci> <ci> c </ci> </list>

Alternatively,
bvar and
condition elements can be used to define lists where membership depends on satisfying certain conditions.
An
order attribute, which is used to specify what ordering is to be used. When the nature of the child elements permits, the ordering defaults to a numeric or lexicographic ordering.
Sets are structured much the same as lists except that there is no implied ordering and the
type of set may be
normal or
multiset with
multiset indicating that repetitions are allowed.
For both sets and lists, the child elements must be valid MathML content elements. The type of the child elements is not restricted. For example, one might construct a list of equations, or inequalities.

matrix and matrixrow

The
matrix element is used to represent mathematical matrices. It is described in detail in
Section 4.4.10.2 [Matrix (matrix)]. It has zero or more child elements, all of which are
matrixrow elements. These in turn expect zero or more child elements that evaluate to algebraic expressions or numbers. These sub-elements are often real numbers, or symbols as in

The
matrixrow elements must always be contained inside of a matrix, and all rows in a given matrix must have the same number of elements.
Note that the behavior of the
matrix and
matrixrow elements is substantially different from the
mtable and
mtr presentation elements.

vector

The
vector element is described in detail in
Section 4.4.10.1 [Vector (vector)]. It constructs vectors from an
n-dimensional vector space so that its
n child elements typically represent real or complex valued scalars as in the three-element vector

The
apply element is described in detail in
Section 4.4.2.1 [Apply (apply)]. Its purpose is apply a function or operator to its arguments to produce an an expression representing an element of the range of the function. It is involved in everything from forming sums such as
a +
b as in

<apply>
<plus/>
<ci> a </ci>
<ci> b </ci>
</apply>

through to using the sine function to construct sin(a) as in

<apply>
<sin/>
<ci> a </ci>
</apply>

or constructing integrals. Its usage in any particular setting is determined largely by the properties of the function (the first child element) and as such its detailed usage is covered together with the functions and operators in
Section 4.2.3 [Functions, Operators and Qualifiers].

indicating an intended comparison between two mathematical values.
MathML 2.0 takes the view that this should be regarded as the application of a boolean function, and as such could be constructed using
apply. The use of
reln with logical operators is supported
for reasons of backwards compatibility, but deprecated in favour of
apply.

fn

The
fn element was used in MathML 1.0 to make
explicit the fact that an expression is being used as a function or
operator. This is allowed in MathML 2.0 for backwards compatibility,
but is deprecated, as the use of
an expression as a function or operator is clear from its position as
the first child of an
apply.
fn is discussed in detail in
Section 4.4.2.3 [Function (fn)].

lambda

The
lambda element is used to construct a user-defined function from an expression and one or more free variables. The lambda construct with
n internal variables takes
n+1 children. The first (second, up to
n) is a
bvar containing the identifiers of the internal variables. The last is an expression defining the function. This is typically an
apply, but can also be any container element. The following constructs
(x, sin
x)

4.2.2.3 Special Constructs

The
declare construct is described in detail in
Section 4.4.2.8 [Declare (declare)]. It is special in that its entire purpose is to modify the semantics of other objects. It is not rendered visually or aurally.

The need for declarations arises any time a symbol (including more general presentations) is being used to represent an instance of an object of a particular type. For example, you may wish to declare that the symbolic identifier
V represents a vector.

The declaration

<declare type="vector"><ci>V</ci></declare>

resets the default type attribute of
<ci>V</ci> to
vector for all affected occurrences of
<ci>V</ci>. This avoids having to write
<ci type="vector">V</ci> every time you use the symbol.

More generally,
declare can be used to associate expressions with specific content. For example, the declaration

associates the symbol
F with a new function defined by the
lambda construct. Within the scope where the declaration is in effect, the expression

<apply>
<ci>F</ci>
<ci> U </ci>
</apply>

stands for the integral of
U from 0 to
a.

The
declare element can also be used to change the definition of a function or operator. For example, if the URL
http://.../MathML:noncommutplus described a non-commutative plus operation encoded in Maple syntax, then the declaration

From the point of view of usage, MathML regards functions (for example
sin and
cos) and operators (for example
plus and
times) in the same way. MathML predefined functions and operators are all canonically empty elements.

Note that the
csymbol element can be used to construct a user-defined symbol that can be used as a function or operator.

4.2.3.1 Predefined functions and operators

MathML functions can be used in two ways. They can be used as the operator within an
apply element, in which case they refer to a function evaluated at a specific value. For example,

<apply>
<sin/>
<cn>5</cn>
</apply>

denotes a real number, namely sin(5).

MathML functions can also be used as arguments to other operators, for example

<apply>
<plus/><sin/><cos/>
</apply>

denotes a function, namely the result of adding the sine and cosine functions in some function space. (The default semantic definition of
plus is such that it infers what kind of operation is intended from the type of its arguments.)

The number of child elements in the
apply is defined by the element in the first (i.e. operator) position.

Unary operators are followed by exactly one other child element within the
apply.

Binary operators are followed by exactly two child elements.

N-ary operators are followed by zero or more child elements.

The one exception to these rules is that
declare elements may be inserted in any position except the first.
declare elements are not counted when satisfying the child element count for an
applycontaining a unary or binary operator element.

Operators taking qualifiers are canonically empty functions that differ from ordinary empty functions only in that they support the use of special
qualifier elements to specify their meaning more fully. They are used in exactly the same way as ordinary operators, except that when they are used as operators, certain qualifier elements are also permitted to be in the enclosing
apply. They always precede the argument if it is present. If more than one qualifier is present, they appear in the order
bvar,
lowlimit,
uplimit,
interval,
condition,
degree,
logbase. A typical example is:

The meaning and usage of qualifier schema varies from function to function. The following list summarizes the usage of qualifier schema with the MathML functions taking qualifiers.

int

The
int function accepts the
lowlimit,
uplimit,
bvar,
interval and
condition schemata. If both
lowlimit and
uplimit schema are present, they denote the limits of a definite integral. The domain of integration may alternatively be specified using interval or condition. The
bvar schema signifies the variable of integration. When used with
int, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.

diff

The
diff function accepts the
bvar schema. The
bvar schema specifies with respect to which variable the derivative is being taken. The
bvar may itself contain a
degree schema that is used to specify the order of the derivative, i.e. a first derivative, a second derivative, etc. For example, the second derivative of
f with respect to
x is:

The
partialdiff function accepts zero or more
bvar schemata. The
bvarschema specify with respect to which variables the derivative is being taken. The
bvar elements may themselves contain
degree schemata that are used to specify the order of the derivative. Variables specified by multiple
bvar elements will be used in order as the variable of differentiation in mixed partials. When used with
partialdiff, the
degree schema is expected to contain a single child schema. For example,

The
sum and
productfunctions accept the
bvar,
lowlimit,
uplimit,
interval and
conditionschemata. If both
lowlimit and
uplimit schemata are present, they denote the limits of the sum or product. The limits may alternatively be specified using the
interval or
conditionschema. The
bvar schema signifies the index variable in the sum or product. A typical example might be:

When used with
sum or
product, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.

limit

The
limit function accepts zero or more
bvar schemata, and optional
condition and
lowlimitschemata. A
condition may be used to place constraints on the
bvar. The
bvar schema denotes the variable with respect to which the limit is being taken. The
lowlimit schema denotes the limit point. When used with
limit, the
bvar and
lowlimit schemata are expected to contain a single child schema; otherwise an error is generated.

log

The
log function accepts only the
logbase schema. If present, the
logbase schema denotes the base with respect to which the logarithm is being taken. Otherwise, the log is assumed to be base 10. When used with
log, the
logbase schema is expected to contain a single child schema; otherwise an error is generated.

moment

The
moment function accepts only the
degree schema. If present, the
degree schema denotes the order of the moment. Otherwise, the moment is assumed to be the first order moment. When used with
moment, the
degree schema is expected to contain a single child schema; otherwise an error is generated.

min, max

The
min and
maxfunctions accept a
bvar schema in cases where the maximum or minimum is being taken over a set of values specified by a
condition schema together with an expression to be evaluated on that set.
In MathML1.0, the
bvar element was optional when using a
condition; if a
condition element containing a single variable was given by itself following a
min or
max operator, the variable was implicitly
assumed to be bound, and the expression to be maximized or minimized
(if absent) was assumed to be the single bound variable. This usage
is deprecated in MathML 2.0 in
favour of explicitly stating the bound variable(s) and the expression
to be maximised in all cases.
The
min and
max elements may also be applied to a list of values in which case no qualifier schemata are used. For examples of all three usages, see
Section 4.4.3.4 [Maximum and minimum (max,
min)].

forall, exists

The universal and existential quantifier operators
forall and
exists are used in conjuction with one or more
bvar schemata to represent simple logical assertions. There are two ways of using the logical quantifier operators. The first usage is for representing a simple, quantified assertion. For example, the statement
`there exists
x< 9' would be represented as:

The MathML content tags include a number of canonically empty elements which denote arithmetic and logical relations. Relations are characterized by the fact that, if an external application were to evaluate them (MathML does not specify how to evaluate expressions), they would typically return a truth value. By contrast, operators generally return a value of the same type as the operands. For example, the result of evaluating
a <
b is either true or false (by contrast, 1 + 2 is again a number).

Relations are bracketed with their arguments using the
apply element in the same way as other functions. In MathML 1.0, relational operators were bracketed using
reln. This usage, although still supported,
is now deprecated in favour of
apply. The element for the relational operator is the first child element of the
apply. Thus, the example from the preceding paragraph is properly marked up as:

<apply>
<lt/>
<ci>a</ci>
<ci>b</ci>
</apply>

It is an error to enclose a relation in an element other than
apply or
reln.

The number of child elements in the
apply is defined by the element in the first (i.e. relation) position.

Unary relations are followed by exactly one other child element within the
apply.

Binary relations are followed by exactly two child elements.

N-ary relations are followed by zero or more child elements.

The one exception to these rules is that
declare elements may be inserted in any position except the first.
declare elements are not counted when satisfying the child element count for an
applycontaining a unary or binary relation element.

4.2.5 Conditions

condition

condition

The
condition element is used to define the
`such that' construct in mathematical expressions. Condition elements are used in a number of contexts in MathML. They are used to construct objects like sets and lists by rule instead of by enumeration. They can be used with the
forall and
exists operators to form logical expressions. And finally, they can be used in various ways in conjunction with certain operators. For example, they can be used with and
int element to specify domains of integration, or to specify argument lists for operators like
min and
max.

The
condition element is always used together with one or more
bvar elements.

The exact interpretation depends on the context, but generally speaking, the
condition element is used to restrict the permissible values of a bound variable appearing in another expression to those that satisfy the relations contained in the
condition. Similarly, when the
condition element contains a
set, the values of the bound variables are restricted to that set.

A condition element contains a single child that is either a
apply, or a
reln element (deprecated). Compound conditions are
indicated by applying relations such as
and inside the child of the condition.

4.2.6 Syntax and Semantics

mappings

semantics,
annotation,
annotation-xml

The use of content markup rather than presentation markup for mathematics is sometimes referred to as
semantic tagging
[Buswell1996]. The parse-tree of a valid element structure using MathML content elements corresponds directly to the expression tree of the underlying mathematical expression. We therefore regard the content tagging itself as encoding the
syntax of the mathematical expression. This is, in general, sufficient to obtain some rendering and even some symbolic manipulation (e.g. polynomial factorization).

However, even in such apparently simple expressions as
X +
Y, some additional information may be required for applications such as computer algebra. Are
X and
Y integers, or functions, etc.?
`Plus' represents addition over which field? This additional information is referred to as
semantic mapping. In MathML, this mapping is provided by the
semantics,
annotation and
annotation-xml elements.

The
semantics element is the container element for the MathML expression together with its semantic mappings.
semantics expects a variable number of child elements. The first is the element (which may itself be a complex element structure) for which this additional semantic information is being defined. The second and subsequent children, if any, are instances of the elements
annotation and/or
annotation-xml.

The
semantics tags also accepts the
definitionURL and
encoding attributes for use by external processing applications. One use might be a URI for a semantic content dictionary, for example. Since the semantic mapping information might in some cases be provided entirely by the
definitionURLattribute, the
annotation or
annotation-xml elements are optional.

The
annotation element is a container for arbitrary data. This data may be in the form of text, computer algebra encodings, C programs, or whatever a processing application expects.
annotation has an attribute
encoding defining the form in use. Note that the content model of
annotation is
PCDATA, so care must be taken that the particular encoding does not conflict with XML parsing rules.

The
annotation-xml element is a container for semantic information in well-formed XML. For example, an XML form of the OpenMath semantics could be given. Another possible use here is to embed, for example, the presentation tag form of a construct given in content tag form in the first child element of
semantics (or vice versa).
annotation-xml has an attribute
encoding defining the form in use.

where
OMA is the element defining the additional semantic information.

Of course, providing an explicit semantic mapping at all is optional, and in general would only be provided where there is some requirement to process or manipulate the underlying mathematics.

4.2.7 Semantic Mappings

Although semantic mappings can easily be provided by various proprietary, or highly specialized encodings, there are no widely available, non-proprietary standard schemes for semantic mapping. In part to address this need, the goal of the OpenMath effort is to provide a platform-independent, vendor-neutral standard for the exchange of mathematical objects between applications. Such mathematical objects include semantic mapping information. The OpenMath group has defined an SGML syntax for the encoding of this information
[OpenMath1996]. This element set could provide the basis of one
annotation-xml element set.

An attractive side of this mechanism is that the OpenMath syntax is specified in XML, so that a MathML expression together with its semantic annotations can be validated using XML parsers.

4.2.8 Constants and Symbols

MathML provdies a collection of predefined constants and symbols which represent frequently-encountered concepts in K-12 mathematics. These include symbols for well-known sets, such as integers
integers and
rationals, and also some widely known constant symbols such as
false,
true,
exponentiale.

4.2.9 MathML element types

MathML functions, operators and relations can all be thought of as mathematical functions if viewed in a sufficiently abstract way. For example, the standard addition operator can be regarded as a function mapping pairs of real numbers to real numbers. Similarly, a relation can be thought of as a function from some space of ordered pairs into the set of values {true, false}. To be mathematically meaningful, the domain and range of a function must be precisely specified. In practical terms, this means that functions only make sense when applied to certain kinds of operands. For example, thinking of the standard addition operator, it makes no sense to speak of
`adding' a set to a function. Since MathML content markup seeks to encode mathematical expressions in a way that can be unambiguously evaluated, it is no surprise that the types of operands is an issue.

MathML specifies the types of arguments in two ways. The first way is by providing precise instructions for processing applications about the kinds of arguments expected by the MathML content elements denoting functions, operators and relations. These operand types are defined in a dictionary of default semantic bindings for content elements, which is given in
Appendix C [Content Element Definitions]. For example, the MathML content dictionary specifies that for real scalar arguments the plus operator is the standard commutative addition operator over a field. The elements
cn has a
type attribute with a default value of
real. Thus some processors will be able to use this information to verify the validity of the indicated operations.

Although MathML specifies the types of arguments for functions, operators and relations, and provides a mechanism for typing arguments, a MathML-compliant processor is not required to do any type checking. In other words, a MathML processor will not generate errors if argument types are incorrect. If the processor is a computer algebra system, it may be unable to evaluate an expression, but no MathML error is generated.

4.3 Content Element Attributes

4.3.1 Content Element Attribute Values

Content element attributes are all of the type
CDATA, that is, any character string will be accepted as valid. In addition, each attribute has a list of predefined values, which a content processor is expected to recognize and process. The reason that the attribute values are not formally restricted to the list of predefined values is to allow for extension. A processor encountering a value (not in the predefined list) which it does not recognize may validly process it as the default value for that attribute.

4.3.2 Attributes Modifying Content Markup Semantics

Each attribute is followed by the elements to which it can be applied.

4.3.2.1 base

cn

indicates numerical base of the number. Predefined values: any numeric string.
The default value is
10

4.3.2.2 closure

4.3.2.3 definitionURL

csymbol, declare, semantics, any operator element

points to an external definition of the semantics of the symbol or construct being declared. The value is a URL or URI that should point to some kind of definition. This definition overrides the MathML default semantics.
At present, MathML does not specify the format in which external semantic definitions should be given. In particular,
there is no requirement that the target of the URI be loadable and parsable.An external definition could, for example, define the semantics in human-readable form.
Ideally, in most situations the definition pointed to by the
definitionURL attribute would be some standard, machine-readable format. However, there are several reasons why MathML does not require such a format.
First, no such format currently exists. There are several projects underway to develop and implement standard semantic encoding formats, most notably the OpenMath effort. But by nature, the development of a comprehensive system of semantic encoding is a very large enterprise, and while much work has been done, much additional work remains. Therefore, even though the
definitionURL is designed and intended for use with a formal semantic encoding language such as OpenMath, it is premature to require any one particular format.
Another reason for leaving the format of the
definitionURL attribute unspecified is that there will always be situations where some non-standard format is preferable. This is particularly true in situations where authors are describing new ideas.
It is anticipated that in the near term, there will be a variety of renderer-dependent implementations of the
definitionURL attribute. For example, a translation tool might simply prompt the user with the specified definition in situations where the proper semantics have been overridden, and in this case, human-readable definitions will be most useful. Other software may utilize OpenMath encodings. Still other software may use proprietary encodings, or look for definitions in any of several formats.
As a consequence, authors need to be aware that there is no guarantee a generic renderer will be able to take advantage of information pointed to by the
definitionURL attribute. Of course, when widely-accepted standardized semantic encodings are available, the definitions pointed to can be replaced without modifying the original document. However, this is likely to be labor intensive.
There is no default value for the
definitionURLattribute, i.e. the semantics are defined within the MathML fragment, and/or by the MathML default semantics.

4.3.2.4 encoding

annotation, annotation-xml, csymbol, semantics, all operator elements

indicates the encoding of the annotation, or in the case of
csymbol ,
semantics and operator elements, the syntax of the target referred to by
definitionURL. Predefined values are
MathML-Presentation,
MathML-Content. Other typical values:
TeX,
OpenMath.
The default value is "", i.e. unspecified.

4.3.2.5 nargs

declare

indicates number of arguments for function declarations. Pre-defined values:
nary, or any numeric string.
The default value is
1

4.3.2.8 scope

The default value is
local.
At present, declarations cannot affect anything outside of the containing
math element. Ideally, one would like to make document-wide declarations by setting the value of the
scope attribute to be
global-document. However, the proper mechanism for document-wide declarations very much depends on details of the way in which XML will be embedded in HTML, future XML style sheet mechanisms, and the underlying Document Object Model.
Since these supporting technologies are still in flux at present, the MathML specification does not include
global-document as a pre-defined value of the
scope attribute. It is anticipated, however, that this issue will be revisited in future revisions of MathML as supporting technologies stabilize. In the near term, MathML implementors that wish to simulate the effect of a document-wide declaration are encouraged to pre-process documents in order to distribute document-wide declarations to each individual
math element in the document.

4.3.2.9 type

cn

indicates type of the number. Predefined values:
integer,
rational,
real,
float,
complex,
complex-polar,
complex-cartesian,
constant.
The default value is
real.
Notes. Each data type implies that the data adheres to certain formating conventions, detailed below. If the data fails to conform to the expected format, an error is generated. Details of the individual formats are:

real

A real number is presented in decimal notation. Decimal notation consists of an optional sign
(`+'or
`-') followed by a string of digits possibly separated into an integer and a fractional part by a
`decimal point'. Some examples are 0.3, 1, and -31.56. If a different
base is specified, then the digits are interpreted as being digits computed to that base.
A real number may also be presented in scientific notation. Such numbers have two parts (a mantissa and an exponent) separated by
`e'. The first part is a real number, while the second part is an integer exponent indicating a power of the base. For example, 12.3e5 represents 12.3 times 10^5.

integer

An integer is represented by an optional sign followed by a string of 1 or more
`digits'. What a
`digit' is depends on the
baseattribute. If
base is present, it specifies the base for the digit encoding, and it specifies it base ten. Thus
base='16' specifies a hex encoding. When
base > 10, letters are added in alphabetical order as digits. The legitimate values for
base are therefore between 2 and 36.

rational

A rational number is two integers separated by
<sep/>. If
base is present, it specifies the base used for the digit encoding of both integers.

complex-cartesian

A complex number is of the form two real point numbers separated by
<sep/>.

complex-polar

A complex number is specified in the form of a magnitude and an angle (in radians). The raw data is in the form of two real numbers separated by
<sep/>.

constant

The
constant type is used to denote named constants. For example, an instance of
<cn type="constant">&pi;</cn>should be interpreted as having the semantics of the mathematical constant Pi. The data for a constant
cn tag may be one of the following common constants:

Symbol

Value

&pi;

The usual
&pi; of trigonometry: approximately 3.141592653...

&ExponentialE; (or
&ee;)

The base for natural logarithms: approximately 2.718281828 ...

&ImaginaryI; (or
&ii;)

Square root of -1

&gamma;

Euler's constant: approximately 0.5772156649...

&infin; (or
&infty;)

Infinity. Proper interpretation varies with context

&true;

the logical constant
true

&false;

the logical constant
false

&NotANumber; (or
&NaN;)

represents the result of an ill-defined floating point division

ci

indicates type of the identifier. Predefined values:
integer,
rational,
real,
float,
complex,
complex-polar,
complex-cartesian,
constant, or the name of any content element. The meaning of the various attribute values is the same as that listed above for the
cn element.
The default value is "", i.e. unspecified.

declare

indicates type of the identifier being declared. Predefined values: any content element name.
The default value is
ci , i.e. a generic identifier

set

indicates type of the set. Predefined values:
normal,
multiset.
multiset indicates that repetitions are allowed.
The default value is
normal.

tendsto

indicates the direction from which the limiting value is approached. Predefined values:
above,
below,
two-sided.
The default value is
above.

4.3.3 Attributes Modifying Content Markup Rendering

4.3.3.1 type

The
type attribute, in addition to conveying semantic information, can be interpreted to provide rendering information. For example in

<ci type="vector">V</ci>

a renderer could display a bold
V for the vector.

4.3.3.2 General Attributes

All content elements support the following general attributes that can be used to modify the rendering of the markup.

Content or semantic tagging goes along with the (frequently implicit) premise that, if you know the semantics, you can always work out a presentation form. When an author's main goal is to mark up re-usable, evaluatable mathematical expressions, the exact rendering of the expression is probably not critical, provided that it is easily understandable. However, when an author's goal is more along the lines of providing enough additional semantic information to make a document more accessible by facilitating better visual rendering, voice rendering, or specialized processing, controlling the exact notation used becomes more of an issue.

MathML elements accept an attribute
other (see
Section 7.2.3 [Attributes for unspecified data]), which can be used to specify things not specifically documented in MathML. On content tags, this attribute can be used by an author to express a
preference between equivalent forms for a particular content element construct, where the selection of the presentation has nothing to do with the semantics. Examples might be

inline or displayed equations

script-style fractions

use of
x with a dot for a derivative over d
x/d
t

Thus, if a particular renderer recognized a display attribute to select between script-style and display-style fractions, an author might write

The information provided in the
other attribute is intended for use by specific renderers or processors, and therefore, the permitted values are determined by the renderer being used. It is legal for a renderer to ignore this information. This might be intentional, in the case of a publisher imposing a house style, or simply because the renderer does not understand them, or is unable to carry them out.

4.4 The Content Markup Elements

This section provides detailed descriptions of the MathML content tags. They are grouped in categories that broadly reflect the area of mathematics from which they come, and also the grouping in the MathML DTD. There is no linguistic difference in MathML between operators and functions. Their separation here and in the DTD is for reasons of historical usage.

When working with the content elements, it can be useful to keep in mind the following.

The role of the content elements is analogous to data entry in a mathematical system. The information that is provided is there to facilitate the successful parsing of an expression as the intended mathematical object by a receiving application.

MathML content elements do not by themselves
`perform' any mathematical evaluations or operations. They do not
`evaluate' in a browser and any
`action' that is ultimately taken on those objects is determined entirely by the receiving mathematical application. For example, editing programs and applications geared to computation for the lower grades would typically leave 3 + 4 as is, whereas computational systems targeting a more advanced audience might evaluate this as 7. Similarly, some computational systems might evaluate sin(0) to 0, whereas others would leave it unevaluated. Yet other computational systems might be unable to deal with pure symbolic expressions sin(x) and may even regard it as a data entry error. None of this has any bearing on the correctness of the original MathML representation. Where evaluation is mentioned at all in the descriptions below, it is merely to help clarify the meaning of the underlying operation.

Apart from the instances where there is an explicit interaction with presentation tagging, there is no required rendering (visual or aural) - only a suggested default. As such, the presentations that are included in this section are merely to help communicate to the reader the intended mathematical meaning by association with the same expression written in a more traditional notation.

4.4.1 Token Elements

4.4.1.1 Number (cn)

Discussion

The cn element is used to specify actual
numerical constants. The content model must provide sufficient information
that a number may be entered as data into a computational system. By
default, it represents a signed real number in base 10. Thus, the content
normally consists of PCDATA restricted to a sign, a string of
decimal digits and possibly a decimal point, or alternatively one of the
predefined symbolic constants such as &pi;.

The cn element uses the attribute type to represent other types of numbers such as, for
example, integer, rational, real or complex, and uses the attribute base to specify the numerical base.

In addition to simple PCDATA, cn
accepts as content PCDATA separated by the (empty) element sep. This determines the different parts needed to
construct a rational or complex-cartesian number.

The cn element may also contain arbitrary
presentation markup in its content (see Chapter 3 [Presentation Markup]) so that its
presentation can be very elaborate.

Alternative input notations for numbers are possible, but must be
explicitly defined by using the definitionURL and
encoding attributes, to refer to a written
specification of how a sequence of real numbers separated by <sep/> should be interpreted.

Default Rendering

By default, a contiguous block of
PCDATA contained in a
cn element should render as if it were wrapped in an
mn presentation element. Similarly, presentation markup contained in a
cn element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were contained wrapped in an
mrow element, with contiguous blocks of
PCDATAwrapped in
mn elements.

However, not all mathematical systems that encounter content based tagging do visual or aural rendering. The receiving applications are free to make use of a number in the manner it normally handles numerical data. Some systems might simplify the rational number 12342/2342342 to 6171/1171171 while pure floating point based systems might approximate this as 0.5269085385e-2. All numbers might be re-expressed in base 10. The role of MathML is simply to record enough information about the mathematical object and its structure so that it may be properly parsed.

The following renderings of the above MathML expressions are included both to help clarify the meaning of the corresponding MathML encoding and as suggestions for authors of rendering applications. In each case, no mathematical evaluation is intended or implied.

12345.7

12345

AB3
16

12342 / 2342342

12.3 + 5 i

Polar( 2 , 3.1415 )

4.4.1.2 Identifier (ci)

Discussion

The
ci element is used to name an identifier in a MathML expression (for example a variable). Such names are used to identify mathematical objects. By default they are assumed to represent complex scalars. The
ci element may contain arbitrary presentation markup in its content (see
Chapter 3 [Presentation Markup]) so that its presentation as a symbol can be very elaborate.

The
ci element uses the
type attribute to specify the type of object that it represents. Valid types include
integer,
rational,
real,
float,
complex,
constant, and more generally, any of the names of the MathML container elements (e.g.
vector) or their type values. The
definitionURL and
encoding attributes can be used to extend the definition of
ci to include other types. For example, a more advanced use might require a
complex-vector.

Examples

<ci> x </ci>

<ci type="vector"> V </ci>

<ci>
<msub>
<mi>x</mi>
<mi>a</mi>
</msub>
</ci>

Default Rendering

If the content of a
ci element is tagged using presentation tags, that presentation is used. If no such tagging is supplied then the
PCDATA content would typically be rendered as if it were the content of an
mi element. A renderer may wish to make use of the value of the type attribute to improve on this. For example, a symbol of type
vector might be rendered using a bold face. Typical renderings of the above symbols are:

4.4.1.3 Externally defined symbol (csymbol)

Discussion

The
csymbol element allows a writer to create an element in MathML whose semantics are externally defined (i.e. not in the core MathML content). The element can then be used in a MathML expression as for example an operator or constant. Attributes are used to give the syntax and location of the external definition of the symbol semantics.

Use of
csymbol for referencing external semantics can be contrasted with use of the
semantics to attach additional information in-line (ie. within the MathML fragment) to a MathML construct. See
Section 4.2.6 [Syntax and Semantics]

Attributes

All attributes are
CDATA:

definitionURL

Pointer to external definition of the semantics of the symbol. MathML does not specify a particular syntax in which this definition should be written.

encoding

Gives the syntax of the definition pointed to by definitionURL. An application can then test the value of this attribute to determine whether it is able to process the target of the
definitionURL. This syntax might be text, or a formal syntax such as OpenMath.

Default Rendering

By default, a contiguous block of
PCDATA contained in a
csymbol element should render as if it were wrapped in an
mo presentation element. Similarly, presentation markup contained in a
csymbol element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were contained wrapped in an
mrowelement, with contiguous blocks of
PCDATA wrapped in
mo elements. The examples above would render by default as

As
csymbol is used to support reference to externally defined semantics, it is a MathML error to have embedded content MathML elements within the
csymbolelement.

4.4.2 Basic Content Elements

4.4.2.1 Apply (apply)

Discussion

The
apply element allows a function or operator to be applied to its arguments. Nearly all expression construction in MathML content markup is carried out by applying operators or functions to arguments. The first child of
apply is the operator, to be applied, with the other child elements as arguments.

The
apply element is conceptually necessary in order to distinguish between a function or operator, and an instance of its use. The expression constructed by applying a function to 0 or more arguments is always an element from the range of the function.

Proper usage depends on the operator that is being applied. For example, the
plus operator may have zero or more arguments. while the
minus operator requires one or two arguments to be properly formed.

If the object being applied as a function is not already one of the elements known to be a function (such as
fn,
sin or
plus) then it is treated as if it were the contents of an
fn element.

Some operators such as
diff and
int make use of
`named' arguments. These special arguments are elements that appear as children of the
apply element and identify
`parameters'such as the variable of differentiation or the domain of integration. These elements are discussed further in
Section 4.2.3.2 [Operators taking Qualifiers].

Examples

<apply>
<factorial/>
<cn>3</cn>
</apply>

<apply>
<plus/>
<cn>3</cn>
<cn>4</cn>
</apply>

<apply>
<sin/>
<ci>x</ci>
</apply>

Default Rendering

A mathematical system that has been passed an
apply element is free to do with it whatever it normally does with such mathematical data. It may be that no rendering is involved (e.g. a syntax validator), or that the
`function application' is evaluated and that only the result is rendered (e.g. sin(0)
0).

When an unevaluated
`function application' is rendered there are a wide variety of appropriate renderings. The choice often depends on the function or operator being applied. Applications of basic operations such as
plus are generally presented using an infix notation while applications of
sinwould use a more traditional functional notation such as sin(x). Consult the default rendering for the operator being applied.

Applications of user-defined functions (see
csymbol,
fn) that are not evaluated by the receiving or rendering application would typically render using a traditional functional notation unless an alternative presentation is specified using the
semantics tag.

4.4.2.2 Relation (reln)

Discussion

The
reln element was used in MathML 1.0 to construct an equation or relation. Relations were constructed in a manner exactly analogous to the use of
apply. This usage is deprecated in MathML 2.0 in favour of the more generally usable
apply.

The first child of
reln is the relational operator, to be applied, with the other child elements acting as arguments. See
Section 4.2.4 [Relations] for further details.

Examples

<reln>
<eq/>
<ci> a </ci>
<ci> b </ci>
</reln>

<reln>
<lt/>
<ci> a </ci>
<ci> b </ci>
</reln>

Default Rendering

4.4.2.3 Function (fn)

Discussion

The
fn element makes explicit the fact that a more general (possibly constructed) MathML object is being used in the same manner as if it were a pre-defined function such as
sin or
plus.

fn has exactly one child element, used to give the name (or presentation form) of the function. When
fn is used as the first child of an apply, the number of following arguments is determined by the contents of the
fn.

In MathML 1.0,
fn was also the primary mechanism used to extend the collection of
`known' mathematical functions. This usage is now deprecated in favour of the more generally applicable
csymbol element. (New functions may also be introduced by using
declare in conjunction with a
lambda expression.)

Examples

Default Rendering

An
fn object is rendered in the same way as its content. A rendering application may add additional adornments such as parentheses to clarify the meaning.

4.4.2.4 Interval (interval)

Discussion

The
interval element is used to represent simple mathematical intervals of the real number line. It takes an attribute
closure, which can take on any of the values
open,
closed,
open-closed, or
closed-open, with a default value of
closed.

More general domains are constructed by using the
condition and
bvar elements to bind free variables to constraints.

The
interval element expects
eithertwo child elements that evaluate to real numbers
or one child element that is a
condition defining the
interval.

Examples

Default Rendering

4.4.2.5 Inverse (inverse)

Discussion

The
inverse element is applied to a function in order to construct a generic expression for the functional inverse of that function. (See also the discussion of
inverse in
Section 4.2.1.5 [The inverse construct]). As with other MathML functions,
inverse may either be applied to arguments, or it may appear alone, in which case it represents an abstract inversion operator acting on other functions.

A typical use of the
inverse element is in an HTML document discussing a number of alternative definitions for a particular function so that there is a need to write and define
f(-1)(x). To associate a particular definition with
f(-1), use the
definitionURL and
encodingattributes.

Default Rendering

The default rendering for a functional inverse makes use of a
parenthesized exponent as in f(-1)(x).

4.4.2.6 Separator (sep)

Discussion

The sep element is to separate PCDATA
into separate tokens for parsing the contents of the various specialized
forms of the cn elements. For example, sep is used when specifying the real and imaginary
parts of a complex number (see Section 4.4.1 [Token Elements]). If it
occurs between MathML elements, it is a MathML error.

Examples

<cn type="complex"> 3 <sep/> 4 </cn>

Default Rendering

4.4.2.7 Condition (condition)

Discussion

The condition element is used to place a
condition on one or more free variables or identifiers. The conditions may
be specified in terms of relations that are to be satisfied by the
variables, including general relationships such as set membership.

It is used to define general sets and lists in situations where the
elements cannot be explicitly enumerated. Condition contains either a
single apply or relnelement; the apply element
is used to construct compound conditions. For example, it is used below to
describe the set of all x such that x < 5. See the
discussion on sets in Section 4.4.6 [Theory of Sets]. See Section 4.2.5 [Conditions] for further details.

Default Rendering

4.4.2.8 Declare (declare)

Discussion

The
declare construct has two primary roles. The first is to change or set the default attribute values for a specific mathematical object. The second is to establish an association between a
`name' and an object. Once a declaration is in effect, the
`name' object acquires the new attribute settings, and (if the second object is present) all the properties of the associated object.

The various attributes of the
declare element assign properties to the object being declared or determine where the declaration is in effect.

By default, the scope of a declaration is
`local' to the surrounding container element. Setting the value of the
scope attribute to
global extends the scope of the declaration to the enclosing
mathelement. As discussed in
Section 4.3.2.8 [scope], MathML contains no provision for making document-wide declarations at present, though it is anticipated that this capability will be added in future revisions of MathML, when supporting technologies become available.
declare takes one or two children. The first child, which is mandatory, is a
ci containing the identifier being declared:

<declare type="vector"> <ci> V </ci> </declare>

The second child, which is optional, is a constructor initialising the variable:

The constructor type and the type of the element declared must agree. For example, if the type attribute of the declaration is
fn, the second child (constructor) must be an element equivalent to an
fn element (This would include actual
fn elements,
lambda elements and any of the defined function in the basic set of content tags.) If no type is specified in the declaration then the type attribute of the declared name is set to the type of the constructor (second child) of the declaration. The type attribute of the declaration can be especially useful in the special case of the second element being a semantic tag.

associates the name
J with a one-variable function defined so that
J(x) = ln
y. (Note that because of the type attribute of the
declare element, the second argument must be something of type
fn, namely a known function like
sin, an
fn construct, or a
lambdaconstruct.)

The
type attribute on the declaration is only necessary if if the type cannot be inferred from the type of the second argument.

Even when a declaration is in effect it is still possible to override attributes values selectively as in
<ci type="integer"> V
</ci>. This capability is needed in order to write statements of the form
`Let
S be a member of
S'.

Default Rendering

Since the
declare construct is not directly rendered, most declarations are likely to be invisible to a reader. However, declarations can produce quite different effects in an application which evaluates or manipulates MathML content. While the declaration

is active the symbol v acquires all the properties of the vector,
and even its dimension and components have meaningful values. This may
affect how v is rendered by some applications, as well as how it
is treated mathematically.

4.4.2.9 Lambda (lambda)

Discussion

The lambda element is used to construct a
user-defined function from an expression and one or more free
variables. The lambda construct with n internal variables takes
n+1 children. The first n children identify the variables
that are used as placeholders in the last child for actual parameter
values. See Section 4.2.2.2 [Constructors] for further details.

Such constructs are often used in conjunction with
declare to construct new functions.

Default Rendering

4.4.2.10 Function composition (compose)

Discussion

The compose element represents the function
composition operator. Note that MathML makes no assumption about the domain
and range of the constituent functions in a composition; the domain of the
resulting composition may be empty.

Default Rendering

4.4.2.11 Identity function (ident)

Discussion

The ident element represents the identity
function. MathML makes no assumption about the function space in which the
identity function resides. That is, proper interpretation of the domain
(and hence range) of the identity function depends on the context in which
it is used.

Examples

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4.4.3 Arithmetic, Algebra and Logic

4.4.3.1 Quotient (quotient)

Discussion

The quotient element is the operator used for
division modulo a particular base. When the quotient operator is applied to integer arguments
a and b, the result is the `quotient of
adivided by b'. That is, quotientreturns the unique integer, q such
that a = qb + r. (In common usage,
q is called the quotient and r is the remainder.)

The quotient element takes the attribute definitionURL and encodingattributes, which can be used override the
default semantics.

Example

<apply>
<quotient/>
<ci> a </ci>
<ci> b </ci>
</apply>

Various mathematical applications will use this data in different ways. Editing applications might choose an image such as shown below, while a computationally based application would evaluate it to 2 when
a=13 and
b=5.

Default Rendering

There is no commonly used notation for this concept. Some possible renderings are

quotient of
a divided by
b

integer part of
a/
b

4.4.3.2 Factorial (factorial)

Discussion

The factorial element is used to construct factorials.

The factorial element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Example

<apply>
<divide/>
<ci> a </ci>
<ci> b </ci>
</apply>

As a MathML expression, this does not evaluate. However, on receiving such an expression, some applications may attempt to evaluate and simplify the value. For example, when
a=5 and
b=2 some mathematical applications may evaluate this to 2.5 while others will treat is as a rational number.

Default Rendering

4.4.3.4 Maximum and minimum (max,
min)

Discussion

The elements
max and
minare used to compare the values of their arguments. They return the maximum and minimum of these values respectively.

The
max and
min elements take the
definitionURL and
encoding attributes that can be used to override the default semantics.

Note that the bound variable must be stated even if it might
be implicit in conventional notation. In MathML1.0, the bound variable
and expression to be evaluated (x) could be omitted in the
example below: this usage is deprecated in MathML2.0 in favour of
explicitly stating the bound variable and expression in all cases:

Example

<apply>
<power/>
<ci> x </ci>
<cn> 3 </cn>
</apply>

If this were evaluated at
x= 5 it would yield 125.

Default Rendering

4.4.3.8 Remainder (rem)

Discussion

The rem element is the operator that returns the
`remainder' of a division modulo a particular base. When the
rem operator is applied to integer arguments
a and b, the result is the `remainder of
adivided by b'. That is, rem returns the unique integer, r such that
a = qb+ r, where r <
q. (In common usage, q is called the quotient and
r is the remainder.)

The
rem element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.

Default Rendering

4.4.3.9 Multiplication (times)

Discussion

times takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.

Example

<apply>
<times/>
<ci> a </ci>
<ci> b </ci>
</apply>

If this were evaluated at
a = 5.5 and
b = 3 it would yield 16.5.

Default Rendering

4.4.3.10 Root (root)

Discussion

The root element is used to construct roots. The
kind of root to be taken is specified by a degreeelement, which should be given as the first child
of the apply element enclosing the root element. Thus, square roots correspond to the case
where degree contains the value 2, cube roots
correspond to 3, and so on. If no degree is
present, a default value of 2 is used.

The root element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Example

Mathematical applications designed for the evaluation of such expressions would evaluate this to
true when
a =
false and
b =
true.

Default Rendering

4.4.3.17 Universal quantifier (forall)

The
forall element represents the universal quantifier of logic. It must used in conjunction with one or more bound variables, an optional
condition element, and an assertion, which may either take the form of an
apply or
reln element.

The
forall element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.

Default Rendering

4.4.3.18 Existential quantifier (exists)

The exists element represents the existential
quantifier of logic. It must used in conjuction with one or more bound
variables, an optional condition element, and an
assertion, which may either take the form of an apply or reln element.

The exists element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Example

Default Rendering

4.4.3.21 Argument (arg)

The arg operator (introduced in MathML 2.0))
gives the `argument' of a complex number, which is the angle
(in radians) it makes with the positive real axis. Real negative numbers
have argument equal to + .

The arg element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Example

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4.4.5 Calculus and Vector Calculus

4.4.5.1 Integral (int)

Discussion

The int element is the operator element for an
integral. The lower limit, upper limit and bound variable are given by
(optional) child elements, lowlimit, uplimit and bvar in the
enclosing apply element. The integrand is also
specified as a child element of the enclosing apply
element.

The domain of integration may alternatively be specified by using an interval element, or by a condition element. In such cases, if a bound variable
of integration is intended, it must be specified explicitly. (The
condition may involve more than one symbol.)

The int element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Default Rendering

4.4.5.2 Differentiation (diff)

Discussion

The diff element is the differentiation operator
element for functions of a single real variable. The bound variable is
given by a bvar element that is a child of the
containing apply element. The bvar elements may also contain a degree element, which specifies the order of the
derivative to be taken.

The diff element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Example

Default Rendering

4.4.5.3 Partial Differentiation (partialdiff)

Discussion

The partialdiff element is the partial
differentiation operator element for functions of several real
variables. The bound variables are given by bvar
elements, which are children of the containing apply element. The bvarelements
may also contain a degree element, which specifies
the order of the partial derivative to be taken in that variable.

The partialdiff element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Example

Default Rendering

4.4.5.4 Lower limit (lowlimit)

Discussion

The lowlimit element is the container element
used to indicate the `lower limit' of an operator using
qualifiers. For example, in an integral, it can be used to specify the
lower limit of integration. Similarly, it is also used to specify the lower
limit of an index for sums and products.

The meaning of the lowlimit element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 [Operators taking Qualifiers].

Example

Default Rendering

The default rendering of the
lowlimit element and its contents depends on the context. In the preceding example, it should be rendered as a subscript to the integral sign:

Consult the descriptions of individual operators that make use of the
lowlimit construct for default renderings.

4.4.5.5 Upper limit (uplimit)

Discussion

The uplimit element is the container element
used to indicate the `upper limit' of an operator using
qualifiers. For example, in an integral, it can be used to specify the
upper limit of integration. Similarly, it is also used to specify the upper
limit of an index for sums and products.

The meaning of the uplimit element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 [Operators taking Qualifiers].

Example

Default Rendering

The default rendering of the uplimit element and
its contents depends on the context. In the preceding example, it should be
rendered as a superscript to the integral sign:

Consult the descriptions of individual operators that make use of the
uplimit construct for default renderings.

4.4.5.6 Bound variable (bvar)

Discussion

The bvar element is the container element for
the `bound variable' of an operation. For example, in an
integral it specifies the variable of integration. In a derivative, it
indicates which variable with respect to which a function is being
differentiated. When the bvar element is used to
quantify a derivative, the bvar element may contain
a child degree element that specifies the order of
the derivative with respect to that variable. The bvarelement is also used for the internal variable in
sums and products and for the bound variable used with the universal and
existential quantifiers forall and exists.

The meaning of the bvar element depends on the
context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 [Operators taking Qualifiers].

Default Rendering

The default rendering of the
bvar element and its contents depends on the context. In the preceding examples, it should be rendered as the
x in the d
x of the integral, and as the
x in the denominator of the derivative symbol:

Note that in the case of the derivative, the default rendering of the
degree child of the
bvar element is as an exponent.

Consult the descriptions of individual operators that make use of the
bvar construct for default renderings.

4.4.5.7 Degree (degree)

Discussion

The degree element is the container element for
the `degree' or `order' of an operation. There
are a number basic mathematical constructs that come in families, such as
derivatives and moments. Rather than introduce special elements for each of
these families, MathML uses a single general construct, the degree element for this concept of
`order'.

The meaning of the
degree element depends on the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking qualifiers, consult
Section 4.2.3.2 [Operators taking Qualifiers].

Examples

Default Rendering

4.4.6.2 List (list)

Discussion

The
list element is the container element that constructs a list of elements. Elements can be defined either by explicitly listing the elements, or by using the
bvar and
condition elements.

Lists differ from sets in that there is an explicit order to the elements. Two orders are supported: lexicographic and numeric. The kind of ordering that should be used is specified by the
order attribute.

Example

Default Rendering

4.4.7 Sequences and Series

4.4.7.1 Sum (sum)

Discussion

The sum element denotes the summation
operator. Upper and lower limits for the sum, and more generally a domains
for the bound variables are specified using uplimit, lowlimit or a condition on the bound variables. The index for the
summation is specified by a bvar element.

The sum element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Default Rendering

4.4.7.2 Product (product)

Discussion

The product element denotes the product
operator. Upper and lower limits for the product, and more generally a
domains for the bound variables are specified using uplimit, lowlimit or a condition on the bound variables. The index for the
product is specified by a bvar element.

The product element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Default Rendering

4.4.7.3 Limit (limit)

Discussion

The limit element represents the operation of
taking a limit of a sequence. The limit point is expressed by specifying a
lowlimit and a bvar, or by
specifying a condition on one or more bound
variables.

The limit element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Default Rendering

4.4.7.4 Tends To (tendsto)

Discussion

The tendsto element is used to express the
relation that a quantity is tending to a specified value.

The tendsto element takes the attributes type to set the direction from which the the limiting
value is approached and the definitionURL and encoding attributes, which can be used to override the
default semantics.

Example

<apply>
<ln/>
<ci> a </ci>
</apply>

If
a =
e this will yield the value 1.

Default Rendering

4.4.8.6 Logarithm (log)

Discussion

The log element is the operator that returns a
logarithm to a given base. The base may be specified using a logbase element, which should be the first element
following log, i.e. the second child of the
containing apply element. If the logbase element is not present, a default base of 10 is
assumed.

The log element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.

Example

Default Rendering

4.4.10 Linear Algebra

4.4.10.1 Vector (vector)

Discussion

vector is the container element for a
vector. The child elements form the components of the vector.

For purposes of interaction with matrices and matrix multiplication,
vectors are regarded as equivalent to a matrix consisting of a single
column, and the transpose of a vector behaves the same as a matrix
consisting of a single row.

Example

Default Rendering

4.4.10.6 Selector (selector)

Discussion

The selector element is the operator for
indexing into vectors matrices and lists. It accepts one or more
arguments. The first argument identifies the vector, matrix or list from
which the selection is taking place, and the second and subsequent
arguments, if any, indicate the kind of selection taking place.

When selector is used with a single argument, it
should be interpreted as giving the sequence of all elements in the list,
vector or matrix given. The ordering of elements in the sequence for a
matrix is understood to be first by column, then by row. That is, for a
matrix ( ai,j), where the indices
denote row and column, the ordering would be a1,1,
a1,2, ... a2,1, a2,2 ... etcetera.

When three arguments are given, the last one is ignored for a list or vector, and in the case of a matrix, the second and third arguments specify the row and column of the selected element.

When two arguments are given, and the first is a vector or list, the second argument specifies an element in the list or vector. When a matrix and only one index
i is specified as in

Default Rendering

None. The information contained in annotations may optionally be used by
a renderer able to process the kind of annotation given.

4.4.11.2 Semantics (semantics)

Discussion

The semantics element is the container element
that associates additional representations with a given MathML
construct. The semantics element has as its first
child the expression being annotated, and the subsequent children are the
annotations. There is no restriction on the kind of annotation that can be
attached using the semantics element. For example, one might give a TEX
encoding, or computer algebra input in an annotation.

The representations that are XML based are enclosed in an annotation-xml element while those representations that
are to be parsed as PCDATA are enclosed in an annotation element.

The semantics element takes the definitionURL and encodingattributes, which can be used to reference an
external source for some or all of the semantic information.

An important purpose of the semantics construct
is to associate specific semantics with a particular presentation, or
additional presentation information with a content construct. The default
rendering of a semantics element is the default
rendering of its first child. When a MathML-presentation annotation is
provided, a MathML renderer may optionally use this information to render
the MathML construct. This would typically be the case when the first child
is a MathML content construct and the annotation is provided to give a
preferred rendering differing from the default for the content
elements.

Default Rendering

The default rendering of a semantics element is
the default rendering of its first child.

4.4.11.3 XML-based annotation (annotation-xml)

Discussion

The annotation-xml container element is used to
contain representations that are XML based. It is always used together with
the semantics element, and takes the attribute encoding to define the encoding being used.